Spanning trails in essentially 4-edge-connected graphs

A connected graph G is essentially 4-edge-connected if for any edge cut X of G with |X|<4, either G−X is connected or at most one component of G−X has edges. In this paper, we introduce a reduction method and investigate the existence of spanning trails in essentially 4-edge-connected graphs. As an application, we prove that if G is 4-edge-connected, then for any edge subset X0⊆E(G) with |X0|≤3 and any distinct edges e,e′∈E(G), G has a spanning (e,e′)-trail containing all edges in X0, which solves a conjecture posed in [W. Luo, Z.-H. Chen, W.-G. Chen, Spanning trails containing given edges, Discrete Math. 306 (2006) 87-98].